[Linear algebra] Zero space of a Matrix

The zero space of matrix A is the set of solutions for AX = 0. Method of zero space:Remove the element of matrix A to obtain the primary and free variables. assign a value to the free variables to obtain a special solution. linearly combine the special solutions to obtain zero space.

Assume that the matrix is as follows:

Perform Gaussian elimination on matrix A to get the upper Triangle Matrix U, and then simplify to get the simplest matrix R:

Since the right side of the equation AX = 0 is a zero vector, only removing the element of matrix A does not affect the solution. Therefore, the augmented matrix is not required, so there are:

From the result of Gaussian elimination, we can see that the rank of matrix A is 2. Column 1 and 3 are the principal component columns and column 2 and column 2 are the free columns. The form transformation is as follows:

From the above formula, we can see that X2 and X4 are free variables, and we can assign values at will. X2 = 0, X4 = 1; x2 = 1, X4 = 0 can get two special solutions respectively (Several free variables have several special solutions.):

Then we willThe zero space of matrix A is obtained by linear combination of two special solutions:

In the preceding section, we describe the method of zero space in the matrix from the numerical solution. The following is an analysis from the formula:In the above example ( Changes the column space without changing the row space or zero space.) To obtain the simplest form of R. After column transformation, we can obtain the following matrix:

We can perform the following deformation on the equation:

We perform the preceding transformation to have a better representation (rows are not converted in the row or column, but we need to remember which column is in the unit matrix I, which column is in the free variable matrix F ):

In this way, we can use the equation to obtain the zero space matrix:

From the above derivation, we can see that each column of the obtained zero space matrix is our special solution (please change the order! The result is the same as the previous one when the second and second rows are exchanged ). Therefore, we can get the simplest R from the elimination method, and then we can directly obtain the zero space matrix. Zero space is the linear combination of column vectors in the zero space matrix.Instead of assigning values to X2 and X4 as before, and then going back to the equation to get two special solutions to get the zero space of the matrix.
Here is another example:

Since R already has a good form, column transformation is not required:

Then, the zero space matrix is obtained by solving the equation:

Note: In Matlab, you can use the rRef (A) and null (A, 'R') commands to obtain the simplest matrix R and zero space matrix X.

[Linear algebra] Zero space of a Matrix